Graph of (e^x)*sin(x)

I need to know about the decreasing and increasing nature of this function. I understand this for positive values of x but I cannot find a rule for large negative numbers. I know the function alternates between increasing and decreasing but don't know over what period this happens. Basically if anyone could give me a fully comprehensive explanation of the nature of this function I would be grateful.

With f(x) = e^x * sin(x), we have

f'(x) = e^x * sin(x) + e^x * cos(x) = e^x * (sin(x) + cos(x)).

Now since e^x > 0 for all reals, f'(x) = 0 when sin(x) = -cos(x), which is the

case when x = 3pi/4 + n*pi. So f(x) has maxima at x = 3pi/4 + n*2pi and

minima at x = 7pi/4 + n*2pi for all integers n, and thus the period is 2pi.

Note that f''(x) = e(x)*sin(x) + e^x*cos(x) + e^x*cos(x) - e^x*sin(x) = 2*e^x*cos(x),

so that there are inflection points where cos(x) = 0 ----> x = pi/2 + n*pi.

Also note that, by the second derivative test, since f''(x) < 0 at x = 3pi/4 + n*2pi

there are maxima of f(x) at these values, and since f''(x) > 0 at x = 7pi/4 + n*2pi

there are minima of f(x) at these values, confirming the above conclusions.